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Braided multitwists

Rodrigo de Pool

TL;DR

The paper characterizes braided multitwists in the mapping class group by showing any braided pair $(\tau_A,\tau_B)$ decomposes as a common multitwist $\tau_C$ times two braided components on disjoint curve sets, with exponents in $\{-1,1\}$. It develops geometric and algebraic intersection bounds (including Ivanov’s general formula) to constrain how curves in the two multitwists can intersect and iterates a symmetry argument via $f=\tau_B\tau_A\tau_B$ to prove orbits of curves have size two. By progressively reducing to the case with no common curves and then to reduced multitwists, the paper proves such reduced braided multitwists must be trivial, yielding a precise classification. The results also inform embeddings of braid groups into Map$(S)$ via multitwists, clarifying when a diagonal decomposition of geometric braid embeddings occurs. The methods blend intersection theory with homology actions to achieve a complete structural description beyond the positive-twist setting.

Abstract

We provide a characterization for multitwists satisfying the braid relation in the mapping class group of an orientable surface.

Braided multitwists

TL;DR

The paper characterizes braided multitwists in the mapping class group by showing any braided pair decomposes as a common multitwist times two braided components on disjoint curve sets, with exponents in . It develops geometric and algebraic intersection bounds (including Ivanov’s general formula) to constrain how curves in the two multitwists can intersect and iterates a symmetry argument via to prove orbits of curves have size two. By progressively reducing to the case with no common curves and then to reduced multitwists, the paper proves such reduced braided multitwists must be trivial, yielding a precise classification. The results also inform embeddings of braid groups into Map via multitwists, clarifying when a diagonal decomposition of geometric braid embeddings occurs. The methods blend intersection theory with homology actions to achieve a complete structural description beyond the positive-twist setting.

Abstract

We provide a characterization for multitwists satisfying the braid relation in the mapping class group of an orientable surface.
Paper Structure (10 sections, 21 theorems, 46 equations, 13 figures, 1 table)

This paper contains 10 sections, 21 theorems, 46 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric intersection formulas
  4. Algebraic intersection formulas
  5. Proof of Main Result
  6. Curves in braided multitwists
  7. Reducing braided multitwists
  8. Action on curves and their homology classes
  9. Orbits have size two
  10. Reduced multitwists are trivial

Key Result

Theorem 1.4

Let $S$ be an orientable surface. If $\tau_A, \tau_B \in \mathop{\mathrm{Map}}\nolimits(S)$ are two braided multitwists, then where $\tau_C$ is a common multitwist, $n_i \in \{-1,1\}$ and the curves $a_1,\dots,a_k,\,b_1,\dots,b_k$ are pairwise disjoint except for $i(a_i,b_i)=1$.

Figures (13)

  • Figure 1: Genus five surface.
  • Figure 2: Curve $a=\bigcup_{i=0}^3 a_i$ and the intersecting curves $c_1,\,c_2,\,c_3$.
  • Figure 3: Planar torus $T_i$ for $i(a_i,b_i)=0$.
  • Figure 4: Planar torus $T_i$ for $i(a_i,b_i)=1$ and $X(a_i,\tau_B)=0$.
  • Figure 5: The planar torus $T_i$ for $i(a_i,b_i)=1$ and $X(a_i,\tau_B)=1.$
  • ...and 8 more figures

Theorems & Definitions (46)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.2: Ivanov's hidden formula
  • Example 2.3
  • Lemma 2.4
  • Remark 3.1
  • Lemma 3.2
  • ...and 36 more